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<Paper uid="C04-1012">
  <Title>Restrictions on Monadic Context-Free Tree Grammars</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"> In this section, some terms, definitions and former results which will be used in the rest of this paper are introduced.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.1 Ranked Alphabets, Trees and Substitution
</SectionTitle>
      <Paragraph position="0"> A ranked alphabet is a finite set of symbols in which each symbol is associated with a natural number, called the rank of a symbol. Let be a ranked alphabet. For n 0, it is defined that n = fa 2 j the rank of a is ng.</Paragraph>
      <Paragraph position="1"> The set T (trees over ) is the smallest set of strings over , parentheses and commas such that (1) 0 T and (2) if 1; 2;::: ; n 2 T and a 2 n for some n 1, then a( 1; 2;::: ; n) 2 T .</Paragraph>
      <Paragraph position="2"> Let be the empty string. Let &amp;quot; be the special symbol that may be contained in 0. The yield of a tree is a function from T into defined as follows. For 2 T , (1) if = a 2 ( 0 f&amp;quot;g), yield( ) = a, (1') if = &amp;quot;, yield( ) = , and (2) if = a( 1; 2;::: ; n) for some a 2 n and 1; 2;::: ; n 2 T , yield( ) = yield( 1) yield( 2) yield( n).</Paragraph>
      <Paragraph position="3"> Let X be the fixed countable set of variables x1;x2;:::. It is defined that X0 = ; and for n 1, Xn = fx1;x2;::: ;xng. x1 is situationally denoted by x. T (Xn) is defined to be T [Xn taking the ranks of elements in X are all 0. For 2 T (Xn) and 1; 2;::: ; n 2 T (X), [ 1; 2;::: ; n] is defined to be the result of substituting each i (1 i n) for the occurences of the variable xi in . A tree 2 T (Xn) is linear if no variable occurs more than once in , and nondeleting if all variables in Xn occur at least once in . The set of all linear trees and all nondeleting trees in T (Xn) are denoted by T (dXne) and T (bXnc), respectively.</Paragraph>
      <Paragraph position="4"> In this papaer, the conventional way of illustrating trees is used. See Figure 1. The tree A(b(a);a;B(E;d)) is illustrated as (1). An arbitrary tree 2 T is illustrated as (2). When the variables of a tree 2 T (X3) occur in the order of x1;x2;x3;x1, the tree is illustrated as (3).</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.2 Context-Free Tree Grammars
</SectionTitle>
      <Paragraph position="0"> The context-free tree grammars (CFTGs) were introduced by W. C. Rounds (1970) as tree generating systems. The definition of CFTGs is a direct generalization of context-free grammars (CFGs).</Paragraph>
      <Paragraph position="1">  A context-free tree grammar (CFTG) is a four-tuple G = (N; ;P;S), where: N and are disjoint ranked alphabets of non-terminals and terminals, respectively.</Paragraph>
      <Paragraph position="2"> P is a finite set of rules of the form A(x1;x2;::: ;xn) ! with n 0, A 2 Nn and 2 TN[ (Xn). For A 2 N0, rules are written as A ! instead of A() ! .</Paragraph>
      <Paragraph position="3"> S, the initial nonterminal, is a distinguished symbol in N0.</Paragraph>
      <Paragraph position="4"> For a CFTG G, the one-step derivation G) is the relation on TN[ TN[ such that for a tree  is in P, then G) 0[ [ 1; 2;::: ; n]]. Figure 2 is an example of a one-step derivation where the rule A(x) ! is applied to the tree = 0[A( 00)] and the tree 0[ [ 00]] is obtained.</Paragraph>
      <Paragraph position="5"> An (n-step) derivation is a finite sequence of trees 0; 1;::: ; n 2 TN[ such that n 0 and  Let G and G0 be CFTGs. G and G0 are equivalent if L(G) = L(G0). G and G0 are weakly equivalent if LS(G) = LS(G0).</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.3 Restrictions on CFTGs
</SectionTitle>
      <Paragraph position="0"> A CFTG G = (N; ;P;S) is monadic if the rank of any nonterminal is 0 or 1, i.e., N = N0 [ N1 and Nn = ; for n 2. G is linear if for any rule A(x1;x2;::: ;xn) ! in P, 2 TN[ (dXne), and nondeleting if for any rule A(x1;x2;::: ;xn) ! in P, 2 TN[ (bXnc).</Paragraph>
      <Paragraph position="1"> A CFTG G = (N; ;P;S) is epsilon-free if for any rule A(x1;x2;::: ;xn) ! in P, the symbol &amp;quot; doesn't occur in .</Paragraph>
      <Paragraph position="2"> When G is monadic, all rules are either of the form A(x) ! with A 2 N1 and 2 TN[ (X1) or of the form B ! with B 2 N0 and 2 TN[ .</Paragraph>
      <Paragraph position="3"> When G is monadic, linear and nondeleting, for any rule A(x) ! with A 2 N1 in P, there exists exactly one occurrence of x in .</Paragraph>
      <Paragraph position="4"> For linear, nondeleting, monadic CFTGs, the following results are known.</Paragraph>
      <Paragraph position="5"> Theorem 2.1 (Fujiyoshi and Kasai, 2000) The class of string languages generated by linear, nondeleting, monadic CFTGs coincides with the class of string languages generated by TAGs.</Paragraph>
      <Paragraph position="6"> Theorem 2.2 (Fujiyoshi and Kasai, 2000) For any linear, nondeleting, monadic CFTG, there exists a weakly equivalent linear, nondeleting, monadic CFTG G = (N; ;P;S) that satisfies the following conditions: For any a 2 , the rank of a is either 0 or 2.</Paragraph>
      <Paragraph position="7"> For each A 2 N0, if A ! is in P, then either = a with a 2 0, or = B(C) with B 2</Paragraph>
      <Paragraph position="9"/>
      <Paragraph position="11"> If a linear, nondeleting, monadic CFTG satisfies the condition of Theorem 2.2, it is said that the grammar is in strong normal form1.</Paragraph>
    </Section>
  </Section>
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